Foundations of Computational Mathematics最新文献_第2页

Optimal Convergence Rates for the Spectrum of the Graph Laplacian on Poisson Point Clouds 泊松点云上图拉普拉斯谱的最优收敛速率

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2025-01-22 DOI: 10.1007/s10208-025-09696-9

Scott Armstrong, Raghavendra Venkatraman

引用次数: 0

Accuracy Controlled Schemes for the Eigenvalue Problem of the Radiative Transfer Equation 辐射传递方程特征值问题的精度控制格式

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2025-01-21 DOI: 10.1007/s10208-025-09694-x

Wolfgang Dahmen, Olga Mula

{"title":"Accuracy Controlled Schemes for the Eigenvalue Problem of the Radiative Transfer Equation","authors":"Wolfgang Dahmen, Olga Mula","doi":"10.1007/s10208-025-09694-x","DOIUrl":"https://doi.org/10.1007/s10208-025-09694-x","url":null,"abstract":"The criticality problem in nuclear engineering asks for the principal eigenpair of a Boltzmann operator describing neutron transport in a reactor core. Being able to reliably design, and control such reactors requires assessing these quantities within quantifiable accuracy tolerances. In this paper, we propose a paradigm that deviates from the common practice of approximately solving the corresponding spectral problem with a fixed, presumably sufficiently fine discretization. Instead, the present approach is based on first contriving iterative schemes, formulated in function space, that are shown to converge at a quantitative rate without assuming any a priori excess regularity properties, and that exploit only properties of the optical parameters in the underlying radiative transfer model. We develop the analytical and numerical tools for approximately realizing each iteration step within judiciously chosen accuracy tolerances, verified by a posteriori estimates, so as to still warrant quantifiable convergence to the exact eigenpair. This is carried out in full first for a Newton scheme. Since this is only locally convergent we analyze in addition the convergence of a power iteration in function space to produce sufficiently accurate initial guesses. Here we have to deal with intrinsic difficulties posed by compact but unsymmetric operators preventing standard arguments used in the finite dimensional case. Our main point is that we can avoid any condition on an initial guess to be already in a small neighborhood of the exact solution. We close with a discussion of remaining intrinsic obstructions to a certifiable numerical implementation, mainly related to not knowing the gap between the principal eigenvalue and the next smaller one in modulus. ","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"20 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142991448","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Conley Index for Multivalued Maps on Finite Topological Spaces 有限拓扑空间上多值映射的Conley索引

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-12-09 DOI: 10.1007/s10208-024-09685-4

Jonathan Barmak, Marian Mrozek, Thomas Wanner

引用次数: 0

Generalized Pseudospectral Shattering and Inverse-Free Matrix Pencil Diagonalization 广义伪谱破碎与无逆矩阵铅笔对角化

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-12-09 DOI: 10.1007/s10208-024-09682-7

James Demmel, Ioana Dumitriu, Ryan Schneider

{"title":"Generalized Pseudospectral Shattering and Inverse-Free Matrix Pencil Diagonalization","authors":"James Demmel, Ioana Dumitriu, Ryan Schneider","doi":"10.1007/s10208-024-09682-7","DOIUrl":"https://doi.org/10.1007/s10208-024-09682-7","url":null,"abstract":"We present a randomized, inverse-free algorithm for producing an approximate diagonalization of any (n times n) matrix pencil (A, B). The bulk of the algorithm rests on a randomized divide-and-conquer eigensolver for the generalized eigenvalue problem originally proposed by Ballard, Demmel and Dumitriu (Technical Report 2010). We demonstrate that this divide-and-conquer approach can be formulated to succeed with high probability provided the input pencil is sufficiently well-behaved, which is accomplished by generalizing the recent pseudospectral shattering work of Banks, Garza-Vargas, Kulkarni and Srivastava (Foundations of Computational Mathematics 2023). In particular, we show that perturbing and scaling (A, B) regularizes its pseudospectra, allowing divide-and-conquer to run over a simple random grid and in turn producing an accurate diagonalization of (A, B) in the backward error sense. The main result of the paper states the existence of a randomized algorithm that with high probability (and in exact arithmetic) produces invertible S, T and diagonal D such that (||A - SDT^{-1}||_2 le varepsilon ) and (||B - ST^{-1}||_2 le varepsilon ) in at most (O left( log ^2 left( frac{n}{varepsilon } right) T_{text {MM}}(n) right) ) operations, where (T_{text {MM}}(n)) is the asymptotic complexity of matrix multiplication. This not only provides a new set of guarantees for highly parallel generalized eigenvalue solvers but also establishes nearly matrix multiplication time as an upper bound on the complexity of inverse-free, exact-arithmetic matrix pencil diagonalization.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"47 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142796989","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Locally-Verifiable Sufficient Conditions for Exactness of the Hierarchical B-spline Discrete de Rham Complex in $$mathbb {R}^n$$ 中层次b样条离散de Rham复合体精确性的局部可验证充分条件 $$mathbb {R}^n$$

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-12-04 DOI: 10.1007/s10208-024-09659-6

Kendrick Shepherd, Deepesh Toshniwal

{"title":"Locally-Verifiable Sufficient Conditions for Exactness of the Hierarchical B-spline Discrete de Rham Complex in $$mathbb {R}^n$$","authors":"Kendrick Shepherd, Deepesh Toshniwal","doi":"10.1007/s10208-024-09659-6","DOIUrl":"https://doi.org/10.1007/s10208-024-09659-6","url":null,"abstract":"Given a domain (Omega subset mathbb {R}^n), the de Rham complex of differential forms arises naturally in the study of problems in electromagnetism and fluid mechanics defined on (Omega ), and its discretization helps build stable numerical methods for such problems. For constructing such stable methods, one critical requirement is ensuring that the discrete subcomplex is cohomologically equivalent to the continuous complex. When (Omega ) is a hypercube, we thus require that the discrete subcomplex be exact. Focusing on such (Omega ), we theoretically analyze the discrete de Rham complex built from hierarchical B-spline differential forms, i.e., the discrete differential forms are smooth splines and support adaptive refinements—these properties are key to enabling accurate and efficient numerical simulations. We provide locally-verifiable sufficient conditions that ensure that the discrete spline complex is exact. Numerical tests are presented to support the theoretical results, and the examples discussed include complexes that satisfy our prescribed conditions as well as those that violate them.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"82 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Constrained and Unconstrained Stable Discrete Minimizations for p-Robust Local Reconstructions in Vertex Patches in the de Rham Complex 德拉姆复数顶点补丁中 p-稳健局部重构的有约束和无约束稳定离散最小化

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-11-25 DOI: 10.1007/s10208-024-09674-7

Théophile Chaumont-Frelet, Martin Vohralík

{"title":"Constrained and Unconstrained Stable Discrete Minimizations for p-Robust Local Reconstructions in Vertex Patches in the de Rham Complex","authors":"Théophile Chaumont-Frelet, Martin Vohralík","doi":"10.1007/s10208-024-09674-7","DOIUrl":"https://doi.org/10.1007/s10208-024-09674-7","url":null,"abstract":"We analyze constrained and unconstrained minimization problems on patches of tetrahedra sharing a common vertex with discontinuous piecewise polynomial data of degree p. We show that the discrete minimizers in the spaces of piecewise polynomials of degree p conforming in the (H^1), ({varvec{H}}(textbf{curl})), or ({varvec{H}}({text {div}})) spaces are as good as the minimizers in these entire (infinite-dimensional) Sobolev spaces, up to a constant that is independent of p. These results are useful in the analysis and design of finite element methods, namely for devising stable local commuting projectors and establishing local-best–global-best equivalences in a priori analysis and in the context of a posteriori error estimation. Unconstrained minimization in (H^1) and constrained minimization in ({varvec{H}}({text {div}})) have been previously treated in the literature. Along with improvement of the results in the (H^1) and ({varvec{H}}({text {div}})) cases, our key contribution is the treatment of the ({varvec{H}}(textbf{curl})) framework. This enables us to cover the whole de Rham diagram in three space dimensions in a single setting.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"113 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142713198","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Tribute to Nick Higham 向尼克-海勒姆致敬

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-11-22 DOI: 10.1007/s10208-024-09680-9

Alan Edelman

引用次数: 0

Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints 近端伽勒金：用于点式约束的结构保留有限元方法

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-11-20 DOI: 10.1007/s10208-024-09681-8

Brendan Keith, Thomas M. Surowiec

{"title":"Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints","authors":"Brendan Keith, Thomas M. Surowiec","doi":"10.1007/s10208-024-09681-8","DOIUrl":"https://doi.org/10.1007/s10208-024-09681-8","url":null,"abstract":"The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop a scalable, mesh-independent algorithm for optimal design with pointwise bound constraints. This paper also introduces the latent variable proximal point (LVPP) algorithm, from which the proximal Galerkin method derives. When analyzing the classical obstacle problem, we discover that the underlying variational inequality can be replaced by a sequence of second-order partial differential equations (PDEs) that are readily discretized and solved with, e.g., the proximal Galerkin method. Throughout this work, we arrive at several contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field, density-based topology optimization. The complete proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. Open-source implementations of our methods accompany this work to facilitate reproduction and broader adoption.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"99 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142678312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Classification of Finite Groups: Recent Developements and Open Problems 有限群的分类：最新发展和未决问题

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-11-12 DOI: 10.1007/s10208-024-09688-1

Bettina Eick

引用次数: 0

Quantitative Convergence of a Discretization of Dynamic Optimal Transport Using the Dual Formulation 使用二元公式对动态优化运输进行离散化的定量收敛

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-11-11 DOI: 10.1007/s10208-024-09686-3

Sadashige Ishida, Hugo Lavenant

引用次数: 0